Time signature

"Common time" redirects here. For the short story, see Common Time.

Simple example of a ³⁴ time signature: here there are three quarter-notes per measure.

The time signature (also known as meter signature) is a notational convention used in Western musical notation to specify how many beats are in each measure and which note value constitutes one beat.

In a musical score, the time signature appears at the beginning of the piece, as a time symbol or stacked numerals (such as or ³⁴) immediately following the key signature (or immediately following the clef if the piece is in C major, A minor, or a modal subset). A mid-score time signature, usually immediately following a barline, indicates a change of meter.

There are various types of time signatures, depending on whether the music follows simple rhythms or involves unusual shifting tempos, including: simple (such as ³⁴ or ⁴⁴), compound (e.g., ⁹⁸ or ¹²⁸), complex (e.g., ⁵⁴ or ⁷⁸), mixed (e.g., ⁵⁸ & ³⁸ or ⁶⁸ or ³⁴), additive (e.g., ³⁺²⁺³⁸, fractional (e.g., ^2½4), irrational meters (e.g., ³¹⁰ or ⁵²⁴), or other meters.

Simple time signatures

Basic time signatures: ⁴⁴, also known as common time (); ²², also known as cut time or cut-common time (); plus ²⁴; ³⁴ & ⁶⁸

Simple time signatures consist of two numerals, one stacked above the other:

• the lower numeral indicates the note value which represents one beat (the "beat unit");
• the upper numeral indicates how many such beats there are in a bar.

For instance, ²⁴ means two quarter-note (crotchet) beats; ³⁸ means three eighth-note (quaver) beats.

The most common simple time signatures are ²⁴, ³⁴, and ⁴⁴.

Notational variations in simple time

A semicircle, or , is sometimes used for ⁴⁴ time, also called common time or imperfect time. The symbol is derived from a broken circle used in music notation from the 14th through 16th centuries, where a full circle represented what today would be written in ³² or ³⁴ time, and was called tempus perfectum (perfect time).^[1] The symbol , a "semicircle" with a vertical line through is also a carry-over from the notational practice of late-Medieval and Renaissance music, where it signified tempus imperfectum diminutum (diminished imperfect time)—more precisely, a doubling of the speed, or proportio dupla, in duple meter.^[2] In modern notation, it is used in place of ²² and is called "alla breve" or, colloquially, "cut time" or "cut common time".

Compound time signatures

Main article: Compound meter (music)

In compound meter, subdivisions of the main beat (the upper number) are split into three, not two, equal parts, so that a dotted note (half again longer than a regular note) becomes the beat unit. Compound time signatures are named as if they were simple time signatures in which the one-third part of the beat unit is the beat, so the top number is commonly 6, 9 or 12 (multiples of 3). The lower number is most commonly an 8 (an eighth-note): as in ⁹⁸ or ¹²⁸.

An example

³⁴: A simple signature, comprising three quarter notes. It has a basic feel of:

one two three (as in a waltz)

Each quarter note might comprise two eighth-notes (quavers) giving a total of six such notes, but it still retains that three-in-a-bar "feel":

one and two and three and

⁶⁸: Theoretically, this can be thought of as the same as the six-quaver form of ³⁴ above with the only difference being that the eighth note is selected as the one-beat unit. But whereas the six quavers in ³⁴ had been in three groups of two, ⁶⁸ is practically understood to mean that they are in two groups of three, with a two-in-a-bar feel:

one and a, two and a

Beat and time

Time signatures indicating two beats per bar (whether it is simple or compound) are called duple time; those with three beats to the bar are triple time. To the ear, a bar may seem like one singular beat. For example, a fast waltz, notated in ³⁴ time, may be described as being "one in a bar". Terms such as quadruple (4), quintuple (5), and so on are also occasionally used.

Actual beat divisions

As mentioned above although indicated by the score as a ³⁴ time the actual beat division used can be the whole bar particularly in faster tempos. Correspondingly in slow tempo the beat indicated by the time signature could in actual performance be divided into smaller units.

Interchangeability, rewriting meters

³⁴ equals ³⁸ time at a different tempo  Play (help·info)

On a formal mathematical level the time signatures of e.g. ³⁴ and ³⁸ are interchangeable; there is a sense in which all simple triple time signatures, be they ³⁸, ³⁴, ³² or anything else, and all compound duple times, such as ⁶⁸, ⁶¹⁶ and so on, are equivalent – a piece in ³⁴ can be easily rewritten in ³⁸ simply by halving the length of the notes. Also other rewritings of time signatures are possible, most commonly a simple time signature with triplets can be translated into a compound meter.

¹²⁸ equals ⁴⁴ time at a different tempo and requires the use of tuplets  Play (help·info)

Although formally interchangeable, for a composer or performing musician the different time signatures can and often have different connotations. Firstly and perhaps most importantly a smaller note value in the beat unit implies a more complex notation which can affect the ease of performance; secondly the beaming affects the choice of actual beat divisions mentioned above - it would for example be more natural to use the quarter note/crotchet as a beat unit in ⁶⁴ or ²² than the eight/quaver in ⁶⁸ or ²⁴; thirdly the time signatures are and have traditionally been associated with different styles of music, e.g. it would seem strange to notate a rock tune in ⁴⁸ or ⁴².

Stress and meter

For all meters, the first beat (the downbeat, ignoring any anacrusis) is usually stressed (though not always, for example in reggae where the offbeats are stressed); in time signatures with four groups in the bar (such as ⁴⁴ and ¹²⁸), the third beat is often also stressed, though to a lesser degree. This gives a regular pattern of stressed and unstressed beats, although notes on the "stressed" beats are not necessarily louder or more important.

Most frequent time signatures

Simple time signatures
44 (quadruple)	common time: widely used in most forms of Western popular music. Most common time signature in rock, blues, country, funk, and pop[3]	Simple quadruple drum pattern: divides four beats into two  Play (help·info)
22 (duple)	alla breve, cut time: used for marches and fast orchestral music. Frequently occurs in musical theater. Sometimes called "in 2", but may be notated in 4.	Simple quadruple drum pattern: divides four beats into two  Play (help·info)
42 (quadruple)	never found in early music (which did not use numeric time signatures), and rare since 1600, although Brahms and other composers used it occasionally.
24 (duple)	used for polkas or marches	Simple duple drum pattern: divides two beats into two.
34 (triple)	used for waltzes, minuets, scherzi, country & western ballads, sometimes used in pop.	Simple triple drum pattern: divides three beats into two  Play (help·info)
38 (triple)	also used for the above, but usually suggests higher tempo or shorter hypermeter.
Compound time signatures
68 (duple)	double jigs, polkas, sega, salegy, tarantella, marches, barcarolles, loures, and some rock music.	Compound duple drum pattern: divides two beats into three  Play (help·info)
98 (triple)	"compound triple time", used in triple ("slip") jigs, otherwise occurring rarely (The Ride of the Valkyries and Tchaikovsky's Fourth Symphony, are familiar examples. Prélude à l'après-midi d'un faune's opening bars are in 98)	Compound triple drum pattern: divides three beats into three  Play (help·info)
128 (quadruple)	also common in slower blues (where it is known as shuffle) and doo-wop; also used more recently in rock music. Can also be heard in some jigs like "The Irish Washerwoman".[citation needed]	Compound quadruple drum pattern: divides four beats into three  Play (help·info)

Complex time signatures

See also: List of musical works in unusual time signatures, Quintuple meter, and Septuple meter

Signatures that do not fit into the usual duple or triple categories are known as "complex", "asymmetric", "irregular", "unusual", or "odd" although these are broad terms, and usually a more specific description is appropriate. Most often these can be recognised by the upper number being 5, 7, or some larger number other than 9 or 12.^{[4][not in citation given]} Although these more complex meters were and are common in some non-Western music, they were rarely used in formal written Western music until the 19th century. The first deliberate quintuple meter pieces were "apparently published in Spain between 1516 and 1520",^[4] although other authorities reckon the Delphic Hymns to Apollo (one by Athenaeus entirely in quintuple meter, the other by Limenius predominantly so), carved on the exterior walls of the Athenian Treasury at Delphi in 128 BC, are probably earlier.^[5] The third Larghetto movement of Chopin's Piano Sonata No. 1 (1828) is an early, but by no means the earliest, example of ⁵⁴ time in solo piano music. Reicha's Fugue 20 from his 36 Fugues, published in 1803, is for piano and is in ⁵⁸. The waltz-like second movement of Tchaikovsky's Pathétique Symphony, often described as a "limping waltz",^[6] is a notable example of ⁵⁴ time in orchestral music. Examples from the 20th century include Holst's "Mars, the Bringer of War", (⁵⁴) from the orchestral suite The Planets, and the ending of Stravinsky's Firebird (⁷⁴).

Examples from the Western popular music tradition include Radiohead's "15 Step" (⁵⁴) and "Paranoid Android" (includes ⁷⁴).^[7] Progressive rock also made frequent use of unusual time signatures, the best known examples being the use of shifting meters in The Beatles' "Strawberry Fields Forever" (1966) and the use of quintuple meter in their "Within You, Without You" (1967).^[8]

The jazz composition "Take Five", written in ⁵⁴ time was one of a number of irregular-meter experiments of The Dave Brubeck Quartet, which also produced compositions in ¹¹⁴ ("Eleven Four"), ⁷⁴ ("Unsquare Dance"), and ⁹⁸ ("Blue Rondo à la Turk"), expressed as ^2+2+2+38, this last being a good example of a work in a signature which, despite appearing to be merely compound triple, is actually more complex.

However, such time signatures are only considered unusual from a Western point of view. In contrast, for example, Bulgarian dances use such meters extensively, including forms with 5, 7, 9, 11, 13, 15, 22, 25 and other numbers of beats per measure. These rhythms are notated as additive rhythms based on simple units, usually 2, 3 and 4 beats, though the notation fails to describe the metric "time bending" taking place; or as compound meters, for example the Bulgarian Sedi Donka, consisting of 25 beats divided 7+7+11, where 7 is subdivided 3+2+2 and 11 is subdivided 2+2+3+2+2 or 4+3+4.^{[citation needed]} See Variants below.

Mixed meters

While time signatures usually express a regular pattern of beat stresses continuing through a piece (or at least a section), sometimes composers place a different time signature at the beginning of each bar, resulting in music with an extremely irregular rhythmic feel. In this case the time signatures are an aid to the performers, and not necessarily an indication of meter. The Promenade from Mussorgsky's Pictures at an Exhibition (1874) is a good example:

Mussorgsky's Pictures at an Exhibition "Promenade"  Play (help·info)

Burt Bacharach's song "Promises, Promises" likewise features a constantly changing meter. The Beatles' All You Need is Love varies from ⁷⁴ to ⁴⁴ in different places.

Igor Stravinsky's The Rite of Spring (1913) is famous for its "savage" rhythms:

In such cases, a common convention followed by some composers (e.g., Olivier Messiaen's La Nativité du Seigneur and Quatuor pour la fin du temps) is simply to omit the time signature. Charles Ives's Concord Sonata has measure bars for the first two measures only, then is entirely unbarred for the rest of the piece. Interestingly, it contains the shortest note written up to then, a "256th" note with six flags marked "play as quickly as you can" above the staff. This same small value had been in use only since 1728, when Georg Philipp Telemann used it in the "Lilliputian Chaconne" from his Intrada, nebst burlesquer Suite (the so-called "Gulliver Suite") for two violins.^[9]

Some pieces have no time signature, as there is no discernible meter. This is commonly known as free time. Sometimes one is provided (usually ⁴⁴) so that the performer finds the piece easier to read, and simply has 'free time' written as a direction. Sometimes the word FREE is written downwards on the staff to indicate the piece is in free time. Erik Satie wrote many compositions which are ostensibly in free time, but actually follow an unstated and unchanging simple time signature throughout. Later composers have made more effective use of this device, writing music which is almost devoid of any discernible regularity of pulse.

If two time signatures alternate repeatedly, sometimes the two signatures will be placed together at the beginning of the piece or section, as in this example, the chorus from the song "America" from West Side Story: in this case, it alternates between ⁶⁸ (in two) in the first measure of each pair and ³⁴ (in three) in the second measure.

Variants

Additive meters

To indicate more complex patterns of stresses, such as additive rhythms, more complex time signatures can be used. For example, the signature

which can be written (3+2+3)/8, means that there are 8 quaver beats in the bar, divided as the first of a group of three eighth notes (quavers) is to be stressed, then the first of a group of two, then first of a group of three again. The stress pattern is usually counted as one-two-three-one-two-one-two-three. This kind of time signature is commonly used to notate folk and non-Western types of music. In classical music, Béla Bartók and Olivier Messiaen have used such time signatures in their works. The first movement of Maurice Ravel's Piano Trio in A Minor is written in ⁸⁸, in which the beats are likewise subdivided into 3 + 2 + 3 to reflect Basque dance rhythms.

Romanian musicologist Constantin Brăiloiu had a special interest in compound time signatures, developed while studying the traditional music of certain regions in his country. While investigating the origins of such unusual meters, he learned that they were even more characteristic of the traditional music of neighboring peoples (e.g., the Bulgarians). He suggested that such timings can be regarded as compounds of simple two-beat and three-beat meters, where an accent falls on every first beat, even though, for example in Bulgarian music, beat lengths of 1, 2, 3, 4 are used in the metric description. In addition, when focusing only on the stressed beats, the simple time signatures themselves will count as beats in a slower, compound time. However, there are two different-length beats in this resulting compound time, a one half-again longer than the short beat (or conversely, the short beat is ²/₃ the value of the long). This type of meter is called aksak (the Turkish word for "limping"), "impeded", "jolting", or "shaking", and is described as an "irregular bichronic rhythm". A certain amount of confusion for Western musicians is inevitable, since a measure they would likely regard as ⁷¹⁶, for example, is a three-beat measure in aksak, with one long and two short beats (with subdivisions of 2+2+3, 2+3+2, or 3+2+2).^[10]

Folk music may make use of metric time bends, so that the proportions of the performed metric beat time lengths differ from the exact proportions indicated by the metric. Depending on playing style of the same meter, the time bend can vary from non-existent to considerable; in the latter case, some musicologists may want to assign a different meter. For example, the Bulgarian tune Eleno Mome is written as 7=2+2+1+2, 13=4+4+2+3, 12=3+4+2+3, but an actual performance (e.g., Smithsonian Eleno Mome) may be closer to 4+4+2+3.5. The Macedonian 3+2+2+3+2 meter is even more complicated, with heavier time bends, and the use of quadruples on the threes; the metric beat time proportions may vary with the speed at which the tune is being played. The Swedish Boda Polska (Polska from the parish Boda) has a typical elongated second beat. In Western classical music, metric time bend is used in the performance of the Viennese Waltz. Most Western music uses metric ratios of 2:1, 3:1, or 4:1 (two-, three- or four-beat time signatures)—in other words, integer ratios which determine all beats to be of equal time length; so relative to that, 3:2 and 4:3 ratios correspond to very distinctive metric rhythm profiles—complex accentuation is used in Western music, but not as a part of the metric accentuation, instead viewed as syncopation.^{[citation needed]}

Brăiloiu borrowed a term from Turkish medieval music theory: aksak (Turkish for crippled). Such compound time signatures fall under the aksak rhythm category that he introduced along with a couple more that should describe the rhythm figures in traditional music.^[11] (Aksak is sometimes spelled as aksaac, because there isn't an exact transliteration from medieval Turkish into the Latin alphabet.)^{[citation needed]} The term Brăiloiu revived had moderate success worldwide, but in Eastern Europe it is still frequently used. However, aksak rhythm figures are to be found not only in a few European countries but on all continents, featuring various combinations of the two and three sequences. Yet the longest were found in Bulgaria; the shortest aksak rhythm figures would be the five-beat timing, comprising a two and a three (which can be also ordered as three followed by two).

Other variants

Some composers have used fractional beats: for example, the time signature ^2½4 appears in Carlos Chávez's Piano Sonata No. 3 (1928) IV, m. 1.

Example of Orff's time signatures

Music educator Carl Orff proposed replacing the lower number of the time signature with the actual note value, as shown at right. This system eliminates the need for compound time signatures (described above), which are confusing to beginners. While this notation has not been adopted by music publishers generally (except in Orff's own compositions), it is used extensively in music education textbooks. Similarly, American composers George Crumb and Joseph Schwantner, among others, have used this system in many of their works.

Another possibility is to extend the barline where a time change is to take place above the top instrument's line in a score and to write the time signature there, and there only, saving the ink and effort that would have been spent writing it in each instrument's staff. Henryk Górecki's Beatus Vir is an example of this. Alternatively, music in a large score sometimes has time signatures written as very long, thin numbers covering the whole height of the score rather than replicating it on each staff; this is an aid to the conductor, who can see signature changes more easily.

Irrational meters

These are time signatures, used for so-called irrational measure lengths,^[12] which have a denominator which is not a power of two (1, 2, 4, 8, 16, 32, etc.). These are "based on beats expressed in terms of fractions of full beats in the prevailing tempo," for example ³¹⁰ or ⁵²⁴.^[12] For example, where ⁴⁴ implies a bar construction of four quarter-parts of a whole note (i.e., four quarter notes), ⁴³ implies a bar construction of four third-parts of it. These signatures are only of utility when juxtaposed with other signatures with varying denominators; a piece written entirely in ⁴³, say, could be more legibly written out in ⁴⁴.

Metric modulation is "a somewhat distant analogy".^[12] It is arguable whether the use of these signatures makes metric relationships clearer or more obscure to the musician; it is always possible to write a passage using non-"irrational" signatures by specifying a relationship between some note length in the previous bar and some other in the succeeding one. Sometimes, successive metric relationships between bars are so convoluted that the pure use of irrational signatures would quickly render the notation extremely hard to penetrate. Good examples, written entirely in conventional signatures with the aid of between-bar specified metric relationships, occur a number of times in John Adams' opera Nixon in China (1987), where the sole use of "irrational" signatures would quickly produce massive numerators and denominators.^{[citation needed]}

Historically, this device has been prefigured wherever composers have written tuplets; for example, a ²⁴ bar consisting of 3 triplet crotchets could arguably more sensibly be written as a bar of ³⁶.^{[citation needed]} Henry Cowell's piano piece "Fabric" (1920) throughout employs separate divisions of the bar (anything from 1 to 9) for the three contrapuntal parts, using a scheme of shaped noteheads to make the differences visually clear, but the pioneering of these signatures is largely due to Brian Ferneyhough, who says that he "find[s] that such 'irrational' measures serve as a useful buffer between local changes of event density and actual changes of base tempo".^[12] Thomas Adès has also made extensive use of them, for example in his piano work "Traced Overhead" (1996), the second movement of which contains, among more conventional meters, bars in such signatures as ²⁶, ⁹¹⁴ and ⁵²⁴. His "Piano Quintet" (2000) makes such extensive use of these, including different lines juxtaposed with varying meters, that an alternate form of notation is not immediately obvious, or arguably desirable.^{[citation needed]} A gradual process of diffusion into less rarefied musical circles seems to be underway, hence for example, John Pickard's work "Eden", commissioned for the 2005 finals of the National Brass Band Championships of Great Britain, which contains bars of ³¹⁰.^{[citation needed]}

Notationally, rather than using Cowell's elaborate series of notehead shapes, the same convention has been invoked as when normal tuplets are written; for example, one beat in ⁴⁵ is written as a normal quarter note, four quarter notes complete the bar, but the whole bar lasts only ⁴/₅ of a reference whole note, and a beat ¹/₅ of one (or ⁴/₅ of a normal quarter note). This is notated in exactly the same way that one would write if one were writing the first four quarter notes of five quintuplet quarter notes.

The term "irrational" is not being used here in its mathematical sense: an irrational number is one that cannot be written as a ratio of whole numbers, which all these signatures obviously are. Nevertheless, the term appears to be established now, although at least one such piece with a truly irrational signature already exists: one of Conlon Nancarrow's "Studies for Player Piano" contains a canon where one part is augmented in the ratio √42:1 (approximately 6.48:1).

Early music usage

Mensural time signatures

In the 15th and 16th centuries, a period in which mensural notation was used, there were four basic "mensuration signs", which determined the proportion between the two main units of rhythm. There were no measure or bar lines in music of this period; these signs, the ancestors of modern time signatures, indicate the ratio of duration between different note values. The relation between the breve and the semibreve was called tempus, and the relation between the semibreve and the minim was called prolatio. Unlike modern notation, the duration ratios between these different values was not always 2:1; it could be either 2:1 or 3:1, and that is what, amongst other things, these mensuration signs indicated. A ratio of 3:1 was called complete, perhaps a reference to the Trinity, and a ratio of 2:1 was called incomplete.

A circle used as a mensuration sign indicated tempus perfectum (a circle being a symbol of completeness), while an incomplete circle, resembling a letter C, indicated tempus imperfectum. Assuming the breve to be a beat, this corresponds to the modern concepts of triple meter and duple meter, respectively. In either case, a dot in the center indicated prolatio perfecta while the absence of such a dot indicated prolatio imperfecta, corresponding to simple meter and compound meter.

A rough equivalence of these signs to modern meters would be:

• corresponds to ⁹⁸ meter;
• corresponds to ³⁴ meter;
• corresponds to ⁶⁸ meter;
• corresponds to ²⁴ meter.

N.B.: in modern compound meters the beat is a dotted note value, such as a dotted quarter, because the ratios of the modern note value hierarchy are always 2:1. Dotted notes were never used in this way in the mensural period; the main beat unit was always a simple (undotted) note value.

Proportions

Another set of signs in mensural notation specified the metric proportions of one section to another, similar to a metric modulation. A few common signs are shown:^[13]

• tempus imperfectum diminutum, 1:2 proportion (twice as fast);
• tempus perfectum diminutum, 1:2 proportion (twice as fast);
• or just proportio tripla, 1:3 proportion (three times as fast, similar to triplets).

Often the ratio was expressed as two numbers, one above the other,^[14] looking similar to a modern time signature, although it could have values such as ⁴³, which a conventional time signature could not.

Some proportional signs were not used consistently from one place or century to another. In addition, certain composers delighted in creating "puzzle" compositions which were intentionally difficult to decipher.

In particular, when the sign was encountered, the tactus (beat) changed from the usual semibreve to the breve, a circumstance called alla breve. This term has been sustained to the present day, and although now it means the beat is a minim (half note), in contradiction to the literal meaning of the phrase, it still indicates that the beat has changed to a longer note value.

See also

• Schaffel
• Tala

References

1. ^ The Academic Manual of the Rudiments of Music. London: A. Weeks & Co. Ltd.. January 1949. pp. 17. 
2. ^ Willi Apel, The Notation of Polyphonic Music 900–1600, fifth edition, revised and with commentary; The Medieval Academy of America Publication no. 38 (Cambridge, Massachusetts: The Medieval Academy of America, 1953): 147–48.
3. ^ Scott Schroedl, Play Drums Today! A Complete Guide to the Basics: Level One (Milwaukee: Hal Leonard Corporation, 2001), p. 42. ISBN 0-634-02185-0.
4. ^ ^{a b} Tim Emmons, Odd Meter Bass: Playing Odd Time Signatures Made Easy (Van Nuys: Alfred Publishing, 2008): 4. ISBN 9780739040812. "What is an 'odd meter'?...A complete definition would begin with the idea of music organized in repeating rhythmic groups of three, five, seven, nine, eleven, thirteen, fifteen, etc."
5. ^ Egert Pöhlmann and Martin L. West, Documents of Ancient Greek Music: The Extant Melodies and Fragments, edited and transcribed with commentary by Egert Pöhlmann and Martin L. West (Oxford: Clarendon Press, 2001): 70–71 and 85. ISBN 0-19-815223-X.
6. ^ http://www.classicalnotes.net/classics/pathetique.html
7. ^ Radiohead (musical group). OK Computer, vocal score with guitar accompaniment and tablature (Essex, England: IMP International Music Publications; Miami, FL: Warner Bros. Publications; Van Nuys, Calif.: Alfred Music Co., Inc., 1997):^{[page needed]}. ISBN 0757991661.
8. ^ Edward Macan, Rocking the Classics: English Progressive Rock and the Counterculture (New York: Oxford University Press, 1997): 48. ISBN 9780195098884.
9. ^ Georg Philipp Telemann, Der getreue Music-Meister (Hamburg: Telemann, 1728): 32.
10. ^ Constantin Brăiloiu, "Le rythme Aksak", Revue de Musicologie 33, nos. 99 and 100 (December 1951): 71–108. Citation on pp. 75–76.
11. ^ Gheorghe Oprea, Folclorul muzical românesc (Bucharest: Ed. Muzicala, 2002),^{[page needed]}. ISBN 973-42-0304-5.
12. ^ ^{a b c d} "Brian Ferneyhough", The Ensemble Sospeso
13. ^ Willi Apel, The Notation of Polyphonic Music 900–1600, fifth edition, revised with commentary; The Medieval Academy of America Publication no. 38 (Cambridge, Massachusetts: The Mediaeval Academy of America, 1953), p. 148.
14. ^ Willi Apel, The Notation of Polyphonic Music 900–1600, fifth edition, revised with commentary; The Medieval Academy of America Publication no. 38 (Cambridge, Massachusetts: The Mediaeval Academy of America, 1953), p. 147.

External links

• Grateful Dead songs with unusual time signatures (Grateful Dead)
• Odd Time Obsessed Internet Radio - dedicated to "odd" meters
• audio clips of many time signatures, and Bounce Metronome Pro - a program that can play all the time signatures mentioned in this article, even the truly irrational ones like ^π4